The restriction of an Eulerian path to an initial segment of the path
forms an Eulerian path on the subgraph consisting of the edges in the
initial segment. (Contributed by Mario Carneiro, 12-Mar-2015.)
(Revised by Mario Carneiro, 3-May-2015.)

The induction step for a vertex degree calculation. If the degree of
in the edge
set is , then adding to
the edge set, where , yields degree as well.
(Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario
Carneiro, 28-Feb-2016.)

The induction step for a vertex degree calculation. If the degree
of in the
edge set is , then adding
to the edge set, where , yields degree .
(Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario
Carneiro, 28-Feb-2016.)

The induction step for a vertex degree calculation. If the degree of
in the edge
set is , then adding to
the edge set, where , yields degree .
(Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario
Carneiro, 28-Feb-2016.)

The Konigsberg Bridge problem. If is the graph on four
vertices , with edges , then
vertices each have
degree three, and has
degree
five, so there are four vertices of odd degree and thus by eupath23311 the
graph cannot have an Eulerian path. (Contributed by Mario Carneiro,
11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)

The property " is
simply normal in base ". A number is
simply normal if each digit occurs in the base-
digit string of with frequency (which is consistent with
the expectation in an infinite random string of numbers selected from
).
(Contributed by Mario Carneiro, 6-Apr-2015.)

If is simply normal,
then the function of
relative density
of in the
digit string converges to , i.e. the set of
occurences of in the digit string has natural density
. (Contributed by Mario Carneiro,
6-Apr-2015.)

Define the Godel-set of universal quantification. Here
corresponds to vN , and represents another
formula, and this
expression is where is the -th
variable, is the code for . Note that this is a
class expression, not a wff. (Contributed by Mario Carneiro,
14-Jul-2013.)

Define the satisfaction predicate. This recursive construction builds up
a function over wff codes and simultaneously defines the set of
assignments to all variables from that makes the coded wff true in
the model , where is interpreted as the binary relation on .
The interpretation of the statement is that for the model , is an
valuation of the variables (v0, v1, etc.) and is a code for a wff using that is true under the assignment . The function is defined by
finite recursion; only operates on wffs of
depth at most , and operates on all wffs.
The coding scheme for the wffs is defined so that

Define the predicate which defines the set of valid Godel formulas. The
parameter defines
the maximum height of the formulas: the set
is all formulas of the form
or
(which in our coding scheme is the set
; see df-sat23332 for the full coding
scheme), and each extra level adds to the complexity of the formulas in
. is the
set of all valid formulas. (Contributed by Mario Carneiro,
14-Jul-2013.)

Define the Godel-set of negation. Here the argument is also a
Godel-set corresponding to smaller formulae. Note that this is a
class
expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)

Define the Godel-set of implication. Here the arguments and
are also Godel-sets corresponding to smaller formulae. Note that this
is a class expression, not a wff. (Contributed by Mario
Carneiro,
14-Jul-2013.)

Define the Godel-set of disjunction. Here the arguments and
are also Godel-sets corresponding to smaller formulae. Note that this
is a class expression, not a wff. (Contributed by Mario
Carneiro,
14-Jul-2013.)

Define the Godel-set of equivalence. Here the arguments and
are also Godel-sets corresponding to smaller formulae. Note that this
is a class expression, not a wff. (Contributed by Mario
Carneiro,
14-Jul-2013.)

Define the Godel-set of equality. Here the arguments
correspond to
vN and vP ,
so
actually means v0 v1 ,
not . Here we use the trick
mentioned in ax-ext2264 to introduce equality as a defined notion in
terms
of . The
expression max
here is a convenient way of getting a dummy variable distinct from
and .
(Contributed by Mario Carneiro, 14-Jul-2013.)

Define the Godel-set of existential quantification. Here
corresponds to vN , and represents another
formula, and this
expression is where is the -th
variable, is the code for . Note that this is a
class expression, not a wff. (Contributed by Mario Carneiro,
14-Jul-2013.)

Define the "proves" relation on a set. A wff is true in a model
if for every valuation , the interpretation of the
wff using the membership relation on is true. (Contributed by
Mario Carneiro, 14-Jul-2013.)

The Godel-set version of the Axiom Scheme of Replacement. Since this is a
scheme and not a single axiom, it manifests as a function on wffs, each
giving rise to a different axiom. (Contributed by Mario Carneiro,
14-Jul-2013.)

The input to this function is a sequence (on ) of homomorphisms
. The resulting structure is the
direct limit of the direct system so defined. This function returns the
pair where is the terminal object and is a
sequence of functions such that and
. (Contributed by Mario Carneiro,
2-Dec-2014.)

The input to this function is a sequence (on ) of structures
and homomorphisms .
The resulting structure is the direct limit of the direct system so
defined, and maintains any structures that were present in the original
objects. TODO: generalize to directed sets? (Contributed by Mario
Carneiro, 2-Dec-2014.)

Define the field extension that augments a field with the root of the
given irreducible polynomial, and extends the norm if one exists and the
extension is unique. (Contributed by Mario Carneiro, 2-Dec-2014.)

Temporary construction for the splitting field of a polynomial. The
inputs are a field and a polynomial that we want to split,
along with a tuple in the same format as the output. The output
is a tuple where is the splitting field and
is an injective homomorphism from the original field .

The function works by repeatedly finding the smallest monic irreducible
factor, and extending the field by that factor using the polyFld
construction. We keep track of a total order in each of the splitting
fields so that we can pick an element definably without needing global
choice. (Contributed by Mario Carneiro, 2-Dec-2014.)

Define the splitting field of a finite collection of polynomials, given
a total ordered base field. The output is a tuple where
is the totally
ordered splitting field and is an injective
homomorphism from the original field . (Contributed by Mario
Carneiro, 2-Dec-2014.)

Define the direct limit of an increasing sequence of fields produced by
pasting together the splitting fields for each sequence of polynomials.
That is, given a ring , a strict order on , and a sequence
of finite sets of polynomials to split,
we construct the direct limit system of field extensions by splitting
one set at a time and passing the resulting construction to
HomLim. (Contributed by Mario Carneiro, 2-Dec-2014.)

Define an equivalence relation on -indexed sequences of integers
such that two sequences are equivalent iff the difference is equivalent
to zero, and a sequence is equivalent to zero iff the sum
is a
multiple of for
every .
(Contributed by Mario Carneiro, 2-Dec-2014.)

There is a unique element of ~Qp
-equivalent to any element of , if the sequences are
zero for sufficiently large negative values; this function selects that
element. (Contributed by Mario Carneiro, 2-Dec-2014.)

Define the completion of the -adic rationals. Here we simply
define it as the splitting field of a dense sequence of polynomials
(using as the -th set the collection of polynomials with degree
less than and
with coefficients ).
Krasner's
lemma will then show that all monic polynomials have splitting fields
isomorphic to a sufficiently close Eisenstein polynomial from the list,
and unramified extensions are generated by the polynomial
, which is
in the list. Thus every finite
extension of Qp is a subfield of this field extension, so it is
algebraically closed. (Contributed by Mario Carneiro, 2-Dec-2014.)